Application of BP Neural Network and Fixed Effect in the Relationship between CSR and Female Directors

­is study is to illustrate the two-way inuencingmechanism of female directors’ gender and corporate social responsibility (CSR) of China’s listed companies. According to an empirical test of 3723 enterprises from 2013 to 2020, it reveals that the presence of female directors—especially female executive directors—contributes to the improvement of CSR. In the stage of back propagation (BP) neural network, the model included the overalls of CSR and macroscopic strategic, fulllment content, and disclosed technology three dimensions. It randomly selects 100 samples, and via BP neural network, it predicts that the RKS-CSR rating can eectively show that high-CSR lever enhances the presence of female directors in Chinese listed companies and promotes gender balance in directors’ team, thus achieving gender balance and CSR two-way sustainable development.


Introduction
e signing of a 2018 California bill requiring at least one female member to be on the board of directors of a state-based public company has reignited the international debate about the value of women directors. With the increasing participation of women in business and society, the business ethics and corporate governance issues of gender diversity on boards of directors are receiving increasing attention [1][2][3].
However, the proportion of female directors remains very low worldwide [4], especially in the Asia Paci c region. According to Deloitte's June 2017 report [5], female makes up only 15% of corporate boards around the world. China is already one of the countries with the highest female employment rate in the world. According to Lai et al. [6] the female labor participation rate in China was about 64%, higher than the world average (50.3%), which was not only higher than other Asian countries, such as India and Japan, but also higher than the United States. However, Chinese female executives and gender ratio were not prominent in relevant international studies.
What is more the proportion of female directors has been increasing very slowly. e proportion of female directors of listed companies in China rose from 9.2% in 1999 to 11.7% in 2010 [7], an increase of only 2.5% in 12 years, which is very low compared with 34.4% in France and 17.9% in the United States. Gender equality has become an increasingly important factor to evaluate an enterprises' sustainable development ability. However, the overall practice of gender equality in Chinese enterprises was still in the "primary stage" [8].
In addition, some scholars [9] argue that CSR may also be used as a competitive advantage. e goal of corporate strategy is to gain a long-term competitive edge that will allow businesses to grow sustainably [10]. Corporate social responsibility may help businesses get low-cost or di erentiated competitive advantages by gaining stakeholder support, developing a positive image in society, and so on, thus resulting in the collaborative and sustainable growth of the organization, society, and environment [11,12]. Nevertheless, over the past 30 years of reform and opening-up, China's economic and social development has made great achievements; however, it has also brought serious social problems, such as resource depletion, environmental pollution, major food safety incidents, product quality issues, labor rights and interest issues, accounting fraud, unfair competition, and so on. e 2018 CSR Development Index averages 34.4 points, and the overall value is still in the startup stage, with more than 40% of enterprises scoring less than 20 points, still in the onlooker stage [13]. As a result, the long-term viability of CSR in these nations must be investigated further [14][15][16].
Specifically, the definition of corporate social responsibility (CSR) is suggested to be "promoting social welfare behaviours beyond the interests and legal requirements of the business" [17]. Many previous studies have analyzed the correlation between female directors and CSR [18,19]. In addition, executive team members' characteristics (age, gender, etc.) are associated with CSR decisions [20], especially, women in executive teams and team heterogeneity have a positive impact on CSR [21,22]. Some studies [23][24][25] found that gender diversity on boards and the presence and proportion of female directors have an impact on CSR. e above studies conclude that female directors have a positive impact on CSR. However, many authors support the positive influence while others such as Shleifer et al. [26] support the opposite and Zhai and Liu [27] show a strong U-shaped association between the proportion of female executives and the level of CSR. is is because the implementation of CSR will lead to an increase in costs and a decrease in profits in the short term, and female risk preference [28] affects their CSR decisions.
Although empirical evidence generally supports the positive influence of female board representatives on CSR, how female directors are related to CSR is still the focus of the corporate governance debate [1,26]. Previous studies believed that [3] whether female directors can effectively promote CSR depends on two key factors: the extent to which female directors influence the board and the extent to which companies are motivated to make full use of female directors [29]. According to agency theory, the independent board members are best suited to carry out the monitoring function of the board in a listed company [30]; however, executive directors are more involved in business decisions. erefore, we hypothesize that a key factor in the impact of female directors on CSR is the power of female directors, which are reflected in the number of female directors on the board and the position role of female directors. Studies in developed countries showed that [31] more than three female directors can have a major impact on CSR. Whether this magic number also applies to developing countries such as China has not been verified yet.
Previous studies have focused on the impact of gender diversity on CSR. However, from the perspective of BP, this study additionally proposed whether improving CSR levels can promote gender balance on the board and increase the number of female directors in the Chinese context. An artificial neural network is a parallel distributed information processing system that consists of a large number of simple processing units interconnected in different ways. ese processing units are also called neurons or neural nodes. It imitates the human brain nervous system, learns a set of mutually corresponding input and output signals, excavates the potential law between the two, and then completes the process of the input signal to calculate the output result according to these laws [32]. Compared with traditional research methods, when dealing with nonlinear complex problems, the artificial intelligence approach overcomes the drawbacks of traditional models which depend on linear function modeling, and improves the accuracy of the model, among which the most widely used is the artificial neural network method, for example, Yang [33], Pendharkar [34], Back [35], Yang [28], Zhang [36] Siriopoulos [37], all used different artificial neural network models to evaluate the financial performance of a company and compare the results with traditional linear models. It is proved that the artificial neural network is superior to the traditional linear method in accuracy.
However, there are many factors in the issues that have not been noticed: how the position of female directors, executive or nonexecutive directors can affect CSR, how many female directors an enterprise needs a minimum to get a real impact on CSR, and whether enterprises which reach a high level of CSR are willing to hire female directors, to achieve a virtuous circle of corporate governance. e answers to these questions are of great theoretical and practical significance for further analyzing and perfecting the causal chain of female directors influencing CSR decision-making.
We pose four research questions to fill up the gaps: (1) are companies with high levels of CSR more willing to hire female directors to achieve gender diversity virtuous circles of corporate governance? (2) Does the involvement of female directors in corporate governance have an impact on CSR? (3) Are there any differences in the impact of different roles of female directors on the boards of directors on CSR? (4) What number of female directors will have a real impact on CSR in the Chinese context?
In this study, we used BP neural network to examine whether the level of CSR exerts influence on the employment of female directors. We investigate whether and under what conditions female directors would influence CSR using the fixed-effect model and panel data to overcome the problem of model bias. e following paragraph shows the presentation sequence of this paper. e theoretical model and assumptions for empirical testing are described in the second part; how we collected the data and created the variables, as well as the empirical approach we used to test our hypotheses are shown in the third part; the findings of the empirical analysis are presented in the fourth part; the final part sums up the findings and explores some fascinating implications as well as their limits.

Female Directors and CSR.
Compared with men, women are more inherently risk-averse [38]. Drawing on upper echelon theory, key majority theory, and social role theory, this study investigates the influence of female directors (FDs) on CSR performance. e social role theory of gender differences [39] shows that gender roles specify assumptions and expectations about attributes held by females and males. Females were often regarded for common traits and behaviors [40], such as relationship-oriented, kind, helpful, caring, and sympathetic to others [41].
is expectation reinforces female values that emphasize voluntary concern for the welfare of others, as well as for the welfare of society and nature [42].
In 1984, Hambrick and Mason [43] defined the upper echelons' perspective. e focus on gender diversity and political background of top management will increase the presence of females in top positions, and a diverse top management layer serves as a key element of CSR performance in Chinese firms [21].
e key majority theory argues that it is only after the number of FDs exceeds a certain level that they will have a meaningful effect on the decision-making of the board of directors and on organizational performance; otherwise, it is still impossible for the voice of FDs to be heard. Furthermore, there is a strong link between the number of FDs and organizational results only when FDs exceed 30% of the total [44]. Whether the same conclusion can be drawn in the Chinese market is also one of the issues studied in this paper.
Social role theory espouses that the style of female leadership is more interpersonal, democratic, and participatory [27]. Meanwhile, Huse and Nielsen [45] proposed that their leadership style can be attributed to the impact of women directors on the board of directors. Boulouta [46] also concluded that women are more likely to conform to gender stereotypes in the operation of the board of directors and to pay more attention to "soft" issues such as CSR. Figure 1 shows the theoretical model of this study.

e Relationship between Female Directors and CSR.
Previous research mostly used artificial neural network models to evaluate the financial performance of a company and compared the results with traditional linear models. It is proved that the artificial neural network is superior to the traditional linear method in accuracy [4][5][6][7][8].
Hypothesis 1. Companies with high levels of CSR tend to be gender-diverse.
In this paper, the influence of FDs on CSR-related issues is discussed based on studies which adopt the upper echelon perspective. Nadeem et al. [25] measured sustainable practices with the Environmental, Social and Governance index (ESG index) and found a strong positive link between the percentage of women on the board of directors and sustainable management practices. Meanwhile, Harjoto and Rossi [23] analyzed the data of 156 Italian listed firms from 2002 to 2014 and found a strong association between the involvement of FDs and CSR. Zhou et al. [19] studied charitable donations as a substitute variable for CSR and showed that listed companies with FDs on the board of directors made more charitable donations than those without FDs. However, the findings of Zhai and Liu [47] show a strong U-shaped association between the proportion of female executives and the level of CSR. Executive directors and nonexecutive directors are the two types of executives in China, based on their positions, whose roles and functions are significantly different in corporate decision-making. Huang and Zhou [48] found that, in contrast to female executive directors, female nonexecutive directors reduce business diversification. Jiang and Long [49] studied 404 listed companies on the main board of Shenzhen Stock Exchange in 2013 and observed that the percentage of female independent directors was strongly negatively associated with corporate value.
Based on this analysis, the existing corporate governance system in China shows that, due to the existence of asymmetry of nonexecutive directors and information in management and the lack of time to perform their duties, the impact of female executive directors on CSR is comparatively important.
Hypothesis 3. Compared to female nonexecutive directors, female executive directors have a positive influence on CSR.

Data and Sample. (I)
e CSR Assessment Index of Rankings CSR Ratings was adopted to assess CSR results. In 2009, Rankins CSR Scores, an authoritative credit rating firm for CSR in China, published its index for the first time, and updated its version in 2012. erefore, this paper takes 2013 as the starting year of the study. e scores were divided into three dimensions: macroscopic strategic (M), fulfilment content (C), and disclosed technology (T). (II) based on research practice, 4,426 samples from 2013 to 2020 were screened according to the following principles: (i) excluding 318 listed companies in the financial and insurance sectors; (ii) excluding 244 listed companies with special trading status such as PT, St, and * St; (iii) excluding 72 companies with less than 2 years of listing (with a one-phase lag); (iv) excluding 69 companies with missing core data. e final remaining samples totaled 3,723. Data on the number of FDs and other control variables were obtained from the CSMAR database. To prevent the potential control of extreme values, all continuous variables were winsorized means at level 1%.
is paper mainly used MATLAB and STATA16 for data processing.

Empirical Analysis CSR to FDs.
A BP neural network is a multilayer feed-forward network that has been trained using the error BP algorithm. It is capable of learning and storing a large number of input-output pattern mapping relationships without disclosing the mathematical equations describing the mapping relationships beforehand. In the previous research from the Web of Science index database, there are only two papers applying the BP neural network to the CSR's field [50,51], the application of BP in the field of economics is still in the initial stage, so this part is an experimental test. e purpose of this experiment is to establish a prediction model to judge the existence of female directors in a company based on four characteristic parameters: RKS rating score, M, C, and T evaluation indexes. e model uses an input-level, hidden-layer, and output-layer network structure with a BP 3-layer structure. e model uses RKS rating scores, M, C, and T to predict whether there is a female director in a company. As shown in Figure 2, there are four neurons in the input layer, eight neurons are concealed, and two neurons are in the output layer. Among them, the activation function of the neurons in the hidden layer adopts the asymmetric Sigmoid function, and the function ex- ). e activation function of the neurons in the output layer adopts the linear function, and the expression is f(x) � x. A total of 3723 sets of data were normalized during the input of the network. BP algorithm is used to alter the mass of the overseas layer and the output layer so that the difference between the calculated output and the real sample output is minimized and eventually, the existence of female directors is predicted more accurately.
Based on the characteristics and advantages of neural network, this paper chooses BP neural network to predict the existence of female directors. Four rating indices, RKS rating scores, M, C, and T are chosen based on the specifications of the model establishment in this paper. e sample interval set of the BP neural network should be [0, 1] and [−1, 1], so that the sample set can be standardized. e normalization formula is x k ′ � x k − x min / x max − x min , and single hidden layer of the network design is adopted, which is divided into two network layers: single hidden layer and output layer, as shown in Figure 3.
In this paper, RKS rating scores, M, C, T four evaluation index are a set of inputs, and the existence of female directors of the company is taken as the output target. If there is an existence, the output is 1, and if not, it is 0. Figure 4 shows the training process of MATLAB for the first model, that is, the four input models. Figure 4 shows the prediction result of the BP neural network. It can be seen from the figures that the experimental model's input layer is 4, the hidden layer is 8, and the output layer is 2. After training, the accuracy of the final prediction result is shown in Figure 5, in which the red circle represents the predicted results, the dot represents the real results, and the coincidence of the two represents the forecast of the existence of a female director is the same as the actual results. A total of 100 companies are randomly selected in the prediction, and the accuracy reaches 74%, which further shows that the RKS rating scores, M, C, T four evaluation indices can effectively represent the existence of female directors, and the model can effectively predict the existence.

Empirical Analysis FDs to CSR
(i) Dependent variables. As the data are positively skewed (right-skewed), when the CSR evaluation index of RKS is adopted to measure CSR performance [22,24,49], so that the logarithm of RKS social responsibility report evaluation index is used to measure CSR performance with reference to Li [52], and the variable data are normally distributed after taking the logarithm. (ii) Independent variables. ree dimensions are used to measure FDs: whether there are FDs, "1" for "yes" and "0" for "no" (FD) [53], the number of FDs (FN) [54], and the proportion of FDs (FR). e women executives are also split into executive directors and nonexecutive directors. e number of female executive directors (FEN) and the percentage of female executive directors (FER) and female nonexecutive directors were determined by the number of female nonexecutive directors (FNEN) and the percentage of female nonexecutive directors (FNER). (iii) Controlled variables: First, variables at the level of corporate characteristics, such as years of listing (Age), company size (Csize) [55], corporate return on assets (Roa), asset-liability ratio (Lev) [10,52,53]. Second, variables at the level of ownership characteristics, such as the shareholding ratio of the largest shareholder (TOP1), the shareholding ratio of the board of directors (DShare) [56], and whether it is a state-owned enterprise (State) [57,58]. ird, variables at the level of corporate governance, such as separation of chairman and general manager (Separa), size of the Board (Board), and the proportion of independent directors (Indepe) [7,60].

Research Models
(i) To test Hypothesis 1, the following econometric model is developed: (ii) To test Hypothesis 2, the following econometric model is developed: Journal of Mathematics e FDNR is determined by the presence or absence of FDs, FN, and FR. As for FTYPE, the FEN and FER are used to quantify female executive directors, whereas FNEN and FNER are used to quantify female nonexecutive directors. Control represented all controlled variables. To avoid possible endogenous problems between the data of FDs, corporate finance, and CSR, the independent variables and controlled variables are all taken from the previous year's data [61]. Table 1 is the definition of variables. Table 2 shows that the average board size, proportion of FDs, and proportion of independent directors by industry from 2013-2018.
As can be seen from     hypothesis "HO: all μ i � 0", that is, this model is a fixed effect model. In this analysis, the Pearson correlation coefficient is used to analyze the correlation between variables. As shown in Table 5, all coefficients that measure the correlation between the substitution variables of FDs are above 0.9, the correlation coefficient between the number and proportion of female executives is 0.930, the correlation coefficient between the number and proportion of female executive directors is 0.969, and the correlation coefficient between the number and proportion of female nonexecutive directors is 0.964, suggesting that there is a high correlation between the substitutive variables of means.

Descriptive Statistics and Correlation Analysis.
ere is no significant correlation among FDNR, FTYPE, and CSR, which indicates that there is interference by other factors on the influence of FDs on CSR [62]. e correlation coefficients between independent variables, controlled variables, and between control variables are all below 0.2 and thus very weak; whereas, generally, a relationship with the coefficient above 0.5 is regarded as strong correlation. erefore, to prevent multicollinearity between variables, the variance inflation factor test is performed.

Hypothesis Testing.
e fixed effect is used to test the principal effect of this paper as the reference regression. e findings of Table 6 show that FDs have a major positive influence on CSR if FD, FN, or FR are regarded as separate variables. e number and percentage of FDs had an important positive effect of 0.01 on CSR and of 0.010 (3.07) and 0.110 on CSR (4.19), respectively, indicating that the number and proportion of FDs participating in board governance have a significant role in promoting CSR. As for the controlled variables, the larger the scale of the enterprise and the board of directors, the higher the shareholding ratio of the largest shareholder and the board of directors, the higher the level of CSR; the higher the ratio of assets and liabilities of the enterprise, the lower the level of CSR; and among enterprises with different property rights systems, state-owned enterprises have a relatively high level of CSR.
Hypothesis 2 is also tested with a fixed effect as the benchmark. e findings of Table 7 indicate that the number and percentage of female executive directors have a significant and positive impact on CSR, with coefficients and T-values of 0.012 (2.32) and 0.145 (3.43), respectively. In comparison, the number and percentage of female nonexecutive directors have a positive but not statistically significant influence on CSR, with coefficients and T-values of 0.006 (0.89) and 0.082 (1.54), respectively. is indicates that the advantageous presence of FDs on CSR is largely due to female executive directors, rather than female nonexecutive directors.

Multicollinearity Test.
To prevent multiple regression models from multicollinearity, the test of variance inflation factor (VIF) is performed. e findings of Tables 8 and 9 indicate that the inflation variance factors in Models 1 and 2 are far smaller than the empiric critical value 10 and that the tolerance is between 0 and 1, suggesting that there is no multicollinearity in each model.

Robustness Test.
According to the key majority theory, only after the number of FDs exceeds a certain level will they have a meaningful effect on the decision-making and organizational success of the board of directors, Studies [31] have pointed out that "three" is the magic number whereby females have a substantial impact on group dynamics [63]. Given the specificity of the Chinese context, if there is more than one FD (FD1), if there are more than two FDs (FD2), and whether there are more than three FDs (FD3), then they are taken as dummy variables to assess FDs and are replaced in Model 1 for research. OLS regression is performed to     Table 10 show that FD1 and FD2 have a positive but not significant impact on CSR performance. FD3 has a positive and significant impact on CSR performance. is is consistent with the key majority theory and the findings of foreign studies; that is, in the Chinese context, it takes more than three FDs to break their "minority status" and enhance their self-confidence and courage to speak out on the board.

Discussion
Corporate social responsibility (CSR) is a critical component of a company's long-term existence and collaborative sustainable growth with society and the environment [64]. It meets the realistic expectations of the entire society and can help businesses increase their competitiveness and reputation. e majority of sustainability studies, however, is undertaken in developed market economies, particularly in the United States and Europe. As a result, prosocial behavior and sustainable development in developing economies are still poorly understood.
In the aspect of the development test, BP neural network is used to predict the existence of female directors. e results showed that the RKS rating score, M, C, T four evaluation indices can effectively represent whether there are female directors in the company or not, and the model can effectively predict the existence of female directors. e higher the score of social responsibility, the more female  Note. * shows that it is relevant at the 0.1 level, * * at the 0.05 level, and * * * at the 0.01 level.   directors may exist, which further verifies that the high social responsibility of enterprises pay more attention to the gender balance in the management and make full use of the advantages of female gender characteristics. We examined the impact of FDs scale and type on CSR through fixed effect. Our results were as follows: (a) generally speaking, China's listed companies lacked FDs, and they were still at a "starting" stage in the fulfilment of social responsibility; (b) the FD's involvement in corporate management had a major positive impact on the CSR, and three or more FDs had a noticeable positive influence on Chinese social responsibility; (c) compared with the supervisory role of female nonexecutive directors, the advisory and executive role of female executive directors showed a more significant impact on CSR.

Managerial Implications.
China is an emerging developing country, and the problem of social responsibility will exert great influence on the sustainable development of enterprises and overall social welfare. It is one of the effective approaches to promote the fulfillment of CSR through the arrangement of board structure, so this study has provided important enlightenment for the improvement of CSR performance through the arrangement of board structure, especially based on gender composition. Meanwhile, enterprises are more willing to hire female directors after the level of CSR is improved, indicating that enterprises with high CSR performance pay more attention to gender balance. e combination of the two can produce a virtuous circle impact on corporate management.

Limitations and Directions for Future Research.
is research has some limitations. First, our sample comprised only Chinese listed companies disclosing social responsibility reports, which may lead to a sample selectivity problem, and listed company samples not disclosing social responsibility were absent. Second, owing to version updating of the RKS-CSR database, the unbalanced panel only covered 6 years. Because the data were missing, we had to reduce the sample scale. When we discovered that data for a specific year were missing, we removed those companies from the sample set; firms with significant omissions or outliers were also removed from that year. Finally, multiple collinearity and endogenous problems were considered, but the test for sample Selection Bias was not conducted due to space limitations. We suggest that researchers should solve the sample selectivity problem with Heckman two-step method in the future. In addition, the application of BP neural network in the field of economics is still in the initial stage, lacking previous research background.

Conclusion
Drawing on the "essential plurality principle" and "social role theory" (1, Ditto), we described research about the relationship between the scale and type of FDs and CSR. Our results provide greater understanding of how FDs influences CSR. We also found that the situation in the Chinese context was the same as that of overseas studies, and 3 is the magic number for women to produce a material impact on group dynamics. Meanwhile, enterprises are more willing to hire female directors after the level of CSR is improved, indicating that enterprises with higher CSR performance pay more attention to gender balance. erefore, we call for a new era of economic and social integration in the world, enterprises should pay more attention to gender equality, which has become a global challenge under today's backdrop. ey need to implement from their own prospective to promote the social goal of gender equality, as well as to dig out the social value and economic value which will be brought by gender equality, so that they will become sustainable development enterprises with responsibility. Paying attention to the mainstream trend of gender equality in global society and communities, and regarding social responsibility standards and relevant laws and regulations of the region, this study reveals the impact of this issue on the enterprise. Gender equality has not only attracted the attention of government departments in many countries but also has become an important factor of local culture shock and community economic development in many enterprises during their overseas operations. It is necessary for enterprises to identify the risks and impacts of this issue on their sustainable operation and it should be incorporated into long-term decisions.
Data Availability e datasets used during the current study are available from the corresponding author on reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.